the graph of the cube root parent function $y = sqrt3{x}$ is translated to form $f(x)$ shown on the graph…

the graph of the cube root parent function $y = sqrt3{x}$ is translated to form $f(x)$ shown on the graph. which equation represents $f(x)$? $f(x)=sqrt3{x + 6}+1$ $f(x)=sqrt3{x - 6}+1$ $f(x)=sqrt3{x + 6}-1$ $f(x)=sqrt3{x - 6}-1$

the graph of the cube root parent function $y = sqrt3{x}$ is translated to form $f(x)$ shown on the graph. which equation represents $f(x)$? $f(x)=sqrt3{x + 6}+1$ $f(x)=sqrt3{x - 6}+1$ $f(x)=sqrt3{x + 6}-1$ $f(x)=sqrt3{x - 6}-1$

Answer

Explanation:

Step1: Recall translation rules

For a cube - root function $y = \sqrt[3]{x}$, a horizontal translation of $h$ units and a vertical translation of $k$ units gives the function $y=\sqrt[3]{x - h}+k$. If the graph is shifted left by $h$ units, $h$ is negative in the formula; if shifted right, $h$ is positive. If shifted up by $k$ units, $k$ is positive; if shifted down, $k$ is negative.

Step2: Identify horizontal shift

The parent function $y = \sqrt[3]{x}$ has an inflection - point at $(0,0)$. The inflection - point of $f(x)$ is at $(- 6,1)$. The $x$ - coordinate of the inflection - point has changed from $0$ to $-6$, which means the graph is shifted left by 6 units. In the formula $y=\sqrt[3]{x - h}+k$, when shifted left by 6 units, $h=-6$, so the function becomes $y = \sqrt[3]{x+6}$.

Step3: Identify vertical shift

The $y$ - coordinate of the inflection - point has changed from $0$ to $1$, which means the graph is shifted up by 1 unit. So $k = 1$. The function $f(x)$ is $y=\sqrt[3]{x + 6}+1$.

Answer:

$f(x)=\sqrt[3]{x + 6}+1$